Elliptical distributions

June 24, 2015 — January 3, 2023

Figure 1

Generalising multivariate Gaussians to anything which has a density function of the form \[ f(x)\propto g((x-\mu )'\Sigma ^{-1}(x-\mu ))\] where \(\mu\) is the mean vector, \(\Sigma\) is a positive definite matrix, and \(g:\mathbb{R}^+\to\mathbb{R}^+\). In fact, we do not need the density function to exist; it’s ok if \(\Sigma\) is positive semi-definite or to allow \(g\) to be a generalised function.

Baby steps, though let us have densities for now. If the mean of such an \(X\sim f\) RV exists, it is \(\mu\), and \(\Sigma\) is proportional to the covariance matrix of \(X\), if such a covariance matrix exists.

I assume they did not invent this idea, but Davison and Ortiz (2019) point out that if you have a least-squares-compatible model, usually it can generalise to any elliptical density, which includes many M-estimator-style robust losses.

2 Elliptical processes

See Aste (2021);Bånkestad et al. (2020).

3 Incoming

4 References

Anderson. 2006. An introduction to multivariate statistical analysis.
Aste. 2021. Stress Testing and Systemic Risk Measures Using Multivariate Conditional Probability.”
Bånkestad, Sjölund, Taghia, et al. 2020. The Elliptical Processes: A Family of Fat-Tailed Stochastic Processes.”
Cambanis, Huang, and Simons. 1981. On the Theory of Elliptically Contoured Distributions.” Journal of Multivariate Analysis.
Chamberlain. 1983. A Characterization of the Distributions That Imply Mean—Variance Utility Functions.” Journal of Economic Theory.
Culan, and Adnet. 2016. Regularized Maximum Likelihood Estimation of Covariance Matrices of Elliptical Distributions.” arXiv:1611.10266 [Math, Stat].
Davison, and Ortiz. 2019. FutureMapping 2: Gaussian Belief Propagation for Spatial AI.” arXiv:1910.14139 [Cs].
Fang, Kai-Tai, Kotz, and Ng. 2017. Symmetric Multivariate and Related Distributions.
Fang, Kaitai, and Zhang. 1990. Generalized Multivariate Analysis.
Gupta, Varga, and Bodnar. 2013. Elliptically Contoured Models in Statistics and Portfolio Theory.
Landsman, and Nešlehová. 2008. Stein’s Lemma for Elliptical Random Vectors.” Journal of Multivariate Analysis.
Landsman, Vanduffel, and Yao. 2013. A Note on Stein’s Lemma for Multivariate Elliptical Distributions.” Journal of Statistical Planning and Inference.
Ley, Babić, and Craens. 2021. Flexible Models for Complex Data with Applications.” Annual Review of Statistics and Its Application.
Markatou, Karlis, and Ding. 2021. Distance-Based Statistical Inference.” Annual Review of Statistics and Its Application.
Ortiz, Evans, and Davison. 2021. A Visual Introduction to Gaussian Belief Propagation.” arXiv:2107.02308 [Cs].
Owen, and Rabinovitch. 1983. On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice.” The Journal of Finance.